A solution must satisfy both equations, so it is a point lying on both lines.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
One solution
Different slopes means the lines cross exactly once.
None
Equal slopes with different intercepts means parallel lines that never meet.
Infinitely many
Equal slopes and equal intercepts means one line drawn twice, so every point works.
The intersection is the solution
Graph both lines; where they cross is the point satisfying both equations. It is the most intuitive method and the one that makes the three-outcome classification obvious at a glance.
Graphing gives approximations
If the solution is not at a lattice point, you cannot read it exactly off a graph. Graphing tells you roughly where the answer is and how many there are; algebra tells you exactly what it is.
Convert to slope-intercept first
Rearranging both equations to y = mx + b makes them quick to graph and makes parallel or identical lines obvious from the coefficients before you draw anything.
When graphing is the right tool
Use it to see the structure, to check an algebraic answer, or when the equations are not linear and algebra is hard. Dismissing graphing as imprecise misses that its purpose is understanding rather than precision.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Dev Classify Three Systems
Dev compares slopes and intercepts before drawing anything.
- Step 1
Slopes 2 and 3 differ, so those lines cross once.
Step 4: Your Turn
Practice makes it stick.
Different Slopes
Problem 1 of 2
Slopes 3 and 5. How many solutions?
Parallel
Problem 2 of 2
Slopes 4 and 4, intercepts 1 and 6. How many solutions?
Count the Solutions
1 of 8
Slopes 2 and 7. How many solutions?
2 of 8
Slopes 5 and 5, intercepts 2 and 2. How many? Use 99 for infinitely many.
3 of 8
Slopes 5 and 5, intercepts 2 and 9. How many solutions?
4 of 8
Lines cross at (3, 8). What is x in the solution?
5 of 8
Sort each pair of lines by how many solutions the system has.
Tap something to move it.
- Empty
- Empty
- Empty
6 of 8
Must a solution satisfy both equations? 1 for yes, 0 for no.
7 of 8
y = x and y = 2x cross at which x?
8 of 8
y = x + 1 and y = x + 1. How many? Use 99 for infinitely many.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Slopes 3 and 3, intercepts 1 and 4. How many solutions?
What You Learned
- The solution of a system is a point on both lines.
- Different slopes cross once; equal slopes are parallel or identical.
- Comparing slopes and intercepts settles the count before any drawing.